Metoda casove diskretizace a parcialni diferencialni

N O V É K N I H Y ( B O O K S ) / K Y B E R N E T I K A - 22 (1986), 5
KAREL REKTORYS
Metoda casove diskretizace a
parcialni diferencialni rovnice
SNTL — Nakladatelstvi technicke literatury,
Praha 1985.
Stran 364; 13 obrazku, 6 tabulek; cena
Kcs 5 0 , - .
The reviewed book by Karel Rektorys,
Professor of Mathematics at the Technical
University in Prague, is the Czech edition
of his book entitled "The Method of Discretization in Time and Partial Differential
Equations" (D. Reidel Publ., Dordrecht—
Boston—London, 1982) The method of
discretization in time (here referred to shortly
as MDT), extending the classical Rothe
method, can be successfully applied to evolution PDEs (partial differential equations)
from both theoretical and numerical points
of view, which is demonstrated in the book
under review. We recall that the Rothe method
consists in semi-discretization in time, while
the spacial variables remain continuous. It
should be remarked here that one-dimensional
problems can be solved by this method directly
on hybrid computers, where one variable
may remain continuous. Then this method,
which becomes very popular in contemporary
hybrid computing, is usually referred to as
CSDT (continuous space — discrete time).
Let us briefly go through the contents
of the book. After an introduction (Chapter 1),
in Chapter 2 the author presents a survey of
variational theory of elliptic PDEs, which is
auxiliary in context of MDT where sequences
of elliptic problems are employed. In fact,
this chapter is a shortened version of author's
previous book "Variacni metody v inzenyrskych problemech a v problemech matematicke
fyziky", (SNTL, Praha 1974, 2nd ed.: Variational Methods in Mathematics, Science and
Engineering, D. Reidel Publ., D o r d r e c h t Boston, 1979). Part I (Chapters 3 — 9), entitled
"Examples", is devoted to numerical practice
in MDT and illustrates applications of the
results stated in Part II (Chapters 10—21)
"Theoretical aspects of MDT".
The "Examples" show how MDT can be
applied to various types of evolution PDE,
both linear and nonlinear, parabolic and
hyperbolic, and also to some less traditional
problems: an integrodifferential parabolic
equation, a problem with an integral condition,
and a rheology problem for a bar. The elliptic
boundary value problems arrising by MDT
are solved exactly, if possible, or again numerically using the well-known Ritz method;
then such approximate solutions are called
Ritz-Rothe approximations. Some results on
numerical experiments and several short
programs written in Basic are presented too.
Each example is completed by references to
the results from Part II concerning existence
of a solution and convergence or error estimates
of an approximate solution. Readers may
find very interesting the study of efficiency
of theoretical error estimates, which are shown
to be particularly realistic if the data of evolution problems vary "slowly" in space variables,
while they may be quite pesimistic if the data
quickly oscillate (it concerns parabolic problems, in hyperbolic problems the situation is
shown to be somewhat different).
Part II demonstrates that MDT can be
readily used for the above outlined type of
evolution PDE to obtain all theoretical
results usually expected, i.e. existence, uniqueness, continuous dependence on data, and
regularity of weak or very weak solutions,
convergence and error estimates for the Rothe
method, as well as convergence for the RitzRothe method (of course, the uniqueness is
proved directly, without using MDT). The
explanation proceeds successively from simpler
to more complicated problems, which makes
it very clear. Thus Chapters 11 and 12 deal
only with a linear parabolic PDE having
homogeneous boundary and initial conditions
and time-independent right hand side. In
Chapter 13 the analysis is extended to nonhomogeneous initial and boundary conditions
(of a Dirichlet type) by introducing the notion
of a very weak solution. Chapter 14 handles
additional errors coming from approximation
of the auxiliary elliptic problems, i.e. the
445
mentioned Ritz-Rothe method, and shows
its convergence. In Chapter 15 the analysis
is further extended to the case when the right
hand side as well as the coefficients depend
Lipschitz continuously on time. In Chapter 16
it is shown that MDT can be used for non­
linear problems as well,namely for a parabolic
problem with a uniformly monotone potential
operator. The following two chapters are
devoted to rather less traditional problems,
namely to an integrodifferential parabolic
equation (with a linear elliptic operator,
homogeneous boudary and initial conditions,
and a nonlinear integral operator) and to
a parabolic problem with an integral condition
(so-called problem of hydratational heat).
It is shown that ' M D T quite easily yields
needed apriori estimates when applied to
these problems in such a way that "unpleasant"
integral terms, depending on the solution
only in the past, are discretized in an explicit
manner, while the elliptic terms are treated,
as typical for the Rothe method, in a fully
implicit manner. Chapters 19 and 20 deal
with a hyperbolic problem, first with homoge­
neous initial conditions, and afterwards with
nonhomogeneous ones, using again the concept
of a very weak solution. Very useful comments
are collected in Chapter 21: Since the author
has confined himself to somewhat special
assumptions to make the explanation clear,
the solutions has got some additional pro­
perties than they usually have (Lipschitz
continuity instead of mere continuity, Looestimates instead of L2 ones, etc.). Further­
more, it is demonstrated how to weaken some
of the assumptions (K-ellipticity, Lipschitz
continuous right hand sides, lime-independent
boundary conditions of the Dirichlet type,
and homogeneous boundary conditions of
the Neumann or Newton type are replaced
respectively by K-coercivity, square integrable
right hand sides, and time-dependent or
nonhomogeneous boundary conditions). It
might have been interesting to mention there
a generalization to spaces of distributions,
like L2(0, T; W2k(G)),
which may be very
useful also in technical practice.
It should be emphasized that most of the
results has been obtained by the author
446
himself or by his students. Although the book
cannot be considered (and it has not been
intended) as a monograph about the evolution PDEs, it is undoubtedly a self-contained
textbook about the technique of MDT, which
has been proved as very powerful and universal
and which finds broad usage among both
engineers and theoretical mathematicians. The
book is written in Rektorys's known bright
and, at the samé time, exact style, which may
seem rather slow for experienced mathema­
ticians, but which will be welcomed by those
who works in applications and by advanced
students.
Tomáš Roubíček
JOHN E. MCNAMARA
Local Area Networks
An introduction to the technology
Digital Press, Burlington, Mass. 1985.
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Dvanáct kapitol knihy je věnováno různým
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Karel Šmuk